Substitutability is a principle in object-oriented programming that states that, in a computer program, if S is a subtype of T, then objects of type T may be replaced with objects of type S (i.e. an object of type T may be substituted with any object of a subtype S) without altering any of the desirable properties of T (correctness, task performed, etc.). More formally, the Liskov substitution principle (LSP) is a particular definition of a subtyping relation, called (strong) behavioral subtyping, that was initially introduced by Barbara Liskov in a 1987 conference keynote address titled Data abstraction and hierarchy. It is a semantic rather than merely syntactic relation because it intends to guarantee semantic interoperability of types in a hierarchy, object types in particular. Barbara Liskov and Jeannette Wing formulated the principle succinctly in a 1994 paper as follows:

Subtype Requirement: Let ϕ(x){\displaystyle \phi (x)} be a property provable about objects x{\displaystyle x} of type T. Then ϕ(y){\displaystyle \phi (y)} should be true for objects y{\displaystyle y} of type S where S is a subtype of T.

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Liskov's notion of a behavioral subtype defines a notion of substitutability for objects; that is, if S is a subtype of T, then objects of type T in a program may be replaced with objects of type S without altering any of the desirable properties of that program (e.g. correctness).

Behavioral subtyping is a stronger notion than typical subtyping of functions defined in type theory, which relies only on the contravariance of argument types and covariance of the return type. Behavioral subtyping is provably undecidable in general: if q is the property "method for xalways terminates", then it is impossible for a program (e.g. a compiler) to verify that it holds true for some subtype S of T, even if q does hold for T. Nonetheless, the principle is useful in reasoning about the design of class hierarchies.

Liskov's principle imposes some standard requirements on signatures that have been adopted in newer object-oriented programming languages (usually at the level of classes rather than types; see nominal vs. structural subtyping for the distinction):

No new exceptions should be thrown by methods of the subtype, except where those exceptions are themselves subtypes of exceptions thrown by the methods of the supertype.

In addition to the signature requirements, the subtype must meet a number of behavioral conditions. These are detailed in a terminology resembling that of design by contract methodology, leading to some restrictions on how contracts can interact with inheritance:

History constraint (the "history rule"). Objects are regarded as being modifiable only through their methods (encapsulation). Because subtypes may introduce methods that are not present in the supertype, the introduction of these methods may allow state changes in the subtype that are not permissible in the supertype. The history constraint prohibits this. It was the novel element introduced by Liskov and Wing. A violation of this constraint can be exemplified by defining a mutable point as a subtype of an immutable point. This is a violation of the history constraint, because in the history of the immutable point, the state is always the same after creation, so it cannot include the history of a mutable point in general. Fields added to the subtype may however be safely modified because they are not observable through the supertype methods. Thus, one can derive a circle with fixed center but mutable radius from immutable point without violating LSP.

The rules on pre- and postconditions are identical to those introduced by Bertrand Meyer in his 1988 book Object-Oriented Software Construction. Both Meyer, and later Pierre America, who was the first to use the term behavioral subtyping, gave proof-theoretic definitions of some behavioral subtyping notions, but their definitions did not take into account aliasing that may occur in programming languages that support references or pointers. Taking aliasing into account was the major improvement made by Liskov and Wing (1994), and a key ingredient is the history constraint. Under the definitions of Meyer and America, a MutablePoint would be a behavioral subtype of ImmutablePoint, whereas LSP forbids this.

A typical example that violates LSP is a Square class that derives from a Rectangle class, assuming getter and setter methods exist for both width and height. The Square class always assumes that the width is equal with the height. If a Square object is used in a context where a Rectangle is expected, unexpected behavior may occur because the dimensions of a Square cannot (or rather should not) be modified independently. This problem cannot be easily fixed: if we can modify the setter methods in the Square class so that they preserve the Square invariant (i.e. keep the dimensions equal), then these methods will weaken (violate) the postconditions for the Rectangle setters, which state that dimensions can be modified independently. Violations of LSP, like this one, may or may not be a problem in practice, depending on the postconditions or invariants that are actually expected by the code that uses classes violating LSP. Mutability is a key issue here. If Square and Rectangle had only getter methods (i.e. they were immutable objects), then no violation of LSP could occur.

A variation that would work is an immutable class RandomSquare that derives from a RandomRectangle class. If the RandomRectangle class creates a rectangle of random size, then a RandomSquare class that creates square of random size would preserve the invariant.

Reinhold Plösch, Contracts, scenarios and prototypes: an integrated approach to high quality software, Springer, 2004, ISBN 3-540-43486-0. Contains a gentler introduction to behavioral subtyping in its various forms in chapter 2.